TY - JOUR

T1 - Plane branches with Newton non-degenerate polars

AU - Hefez, A.

AU - Hernandes, M. E.

AU - Iglesias, M. F.H.

N1 - Publisher Copyright:
© 2018 World Scientific Publishing Company.

PY - 2018/1/1

Y1 - 2018/1/1

N2 - To an equisingularity class of complex plane branches, described by its multiplicity n and characteristic exponents β1 < ⋯ < βr, 0 < n < β1, there is a naturally associated family K(n,β1,...,βr) of equations containing a complete set of analytic representatives for all branches of the class. We show in this paper that the general polar curve of any member of K(n,β1,...,βr) is Newton degenerate, except when r = 1, in which case the general member of K(n,β1) corresponds to a curve which has a Newton non-degenerate general polar curve with a fixed Newton polygon, or when r = 2, n = 2p, β1 = 2q, β2 = 2q + d, with GCD(p,q) = 1 and d ≥ 1 is odd, in which case K(2p, 2q, 2q + d) has a subset containing a complete set of analytic representatives for all branches of the class whose general member has also a Newton non-degenerate general polar curve with a fixed Newton polygon. In both cases, we give explicit Zariski open sets the points of which represent branches with Newton non-degenerate polars and describe the topology of their general polars.

AB - To an equisingularity class of complex plane branches, described by its multiplicity n and characteristic exponents β1 < ⋯ < βr, 0 < n < β1, there is a naturally associated family K(n,β1,...,βr) of equations containing a complete set of analytic representatives for all branches of the class. We show in this paper that the general polar curve of any member of K(n,β1,...,βr) is Newton degenerate, except when r = 1, in which case the general member of K(n,β1) corresponds to a curve which has a Newton non-degenerate general polar curve with a fixed Newton polygon, or when r = 2, n = 2p, β1 = 2q, β2 = 2q + d, with GCD(p,q) = 1 and d ≥ 1 is odd, in which case K(2p, 2q, 2q + d) has a subset containing a complete set of analytic representatives for all branches of the class whose general member has also a Newton non-degenerate general polar curve with a fixed Newton polygon. In both cases, we give explicit Zariski open sets the points of which represent branches with Newton non-degenerate polars and describe the topology of their general polars.

KW - analytic equivalence

KW - equisingularity

KW - Polar curves

UR - http://www.scopus.com/inward/record.url?scp=85040953836&partnerID=8YFLogxK

U2 - 10.1142/S0129167X18500015

DO - 10.1142/S0129167X18500015

M3 - Article

AN - SCOPUS:85040953836

SN - 0129-167X

VL - 29

JO - International Journal of Mathematics

JF - International Journal of Mathematics

IS - 1

M1 - 1850001

ER -